aa r X i v : . [ m a t h . DG ] A p r THE ANOMALY FLOW ON NILMANIFOLDS
MATTIA PUJIA AND LUIS UGARTE
Abstract.
We study the Anomaly flow on -step nilmanifolds with respect to any Hermitianconnection in the Gauduchon line. In the case of flat holomorphic bundle, the general solution tothe Anomaly flow is given for any initial invariant Hermitian metric. The solutions depend on twoconstants K and K , and we study the qualitative behaviour of the Anomaly flow in terms oftheir signs, as well as the convergence in Gromov-Hausdorff topology. The sign of K is related tothe conformal invariant introduced by Fu, Wang and Wu. In the non-flat case, we find the generalevolution equations of the Anomaly flow under certain initial assumptions. This allows us to detectnon-flat solutions to the Hull-Strominger-Ivanov system on a concrete nilmanifold, which appearas stationary points of the Anomaly flow with respect to the Bismut connection. Contents
1. Introduction 12. Preliminaries 42.1. Adapted bases 52.2. Trace of the curvature 73. The first Anomaly flow equation on nilpotent Lie groups 103.1. Special Hermitian metrics along the flow 123.2. Reduction to almost diagonal initial metrics and the general solution 134. The Anomaly flow with flat holomorphic bundle 154.1. Qualitative behaviour of the model problem 154.2. The sign of K and its relation to the Fu-Wang-Wu conformal invariant 184.3. Convergence of the nilmanifolds 195. Evolution of the holomorphic vector bundle 205.1. Anomaly flow on N and solutions to the Hull-Strominger-Ivanov system 236. Appendix A 277. Appendix B 29References 321. Introduction
The
Anomaly flow is a coupled flow of Hermitian metrics introduced by Phong, Picard and Zhangin [30]. The flow, which was originally proposed as a new tool to detect explicit solutions to the
Mathematics Subject Classification.
Primary 53C44 ; Secondary 53C15, 53B15, 53C30.
Key words and phrases.
Anomaly flow; Hull-Strominger system; Nilmanifolds.The first-named author was partially supported by G.N.S.A.G.A. of I.N.d.A.M. The second-named author wassupported by the projects MTM2017-85649-P (AEI/FEDER, UE), and E22-20R “Álgebra y Geometría” (Gobiernode Aragón/FEDER).
Hull-Strominger system, leads to some interesting problems in complex non-Kähler geometry andits study is just began [7, 9, 10, 31, 32, 33, 34].Let X be a 3-dimensional complex manifold equipped with a nowhere vanishing (3 , -form Ψ anda holomorphic vector bundle π : E → X . In this paper we consider the coupled flow of Hermitianmetrics ( ω t , H t ) , with ω t on X and H t along the fibers of E , given by ∂ t ( k Ψ k ω t ω t ) = i∂∂ω t − α ′ Rm τt ∧ Rm τt ) − Tr( A κt ∧ A κt )) ,H − t ∂ t H t = ω t ∧ A κt ω t , (1)where Rm τ and A κ are, respectively, the curvature tensors of Gauduchon connections ∇ τ on ( X, ω t ) and ∇ κ on ( E, H t ) , and α ′ ∈ R is the so-called slope parameter .Phong, Picard and Zhang proved that, if the connections ∇ τ and ∇ κ in (1) are both Chern,then the flow preserves the conformally balanced condition d ( k Ψ k ω ω ) = 0 and, under an extraassumption on the initial metric ω , it is well-posed [30]. If furthermore ω is conformally balancedand the flow is defined for every t ∈ [0 , ∞ ) , then its limit points ( ω ∞ , H ∞ ) are automaticallysolutions to the Hull-Strominger system [30].More generally, any stationary solution to the Anomaly flow (1), satisfying the conformallybalanced condition and for which the curvature form A κ is of type (1 , , is a solution to the Hull-Strominger system [22, 41]: ω ∧ A κ = 0 , ( A κ ) , = ( A κ ) , = 0 ,i ∂∂ω = α ′ Rm τ ∧ Rm τ ) − Tr( A κ ∧ A κ )) ,d ( k Ψ k ω ω ) = 0 . (2)Here, the first two equations represent the Hermitian-Yang-Mills equation for the connection ∇ κ ;the third equation follows by the Green-Schwarz cancellation mechanism in string theory and it isknown as anomaly cancellation ; while, the last equation was originally formulated as d ∗ ω = i ( ¯ ∂ − ∂ ) ln k Ψ k ω , where d ∗ is the co-differential, and the above equivalent expression is due to Li and Yau [27].We mention that the Hull-Strominger system arises from the symmetric compactification of the10-dimensional heterotic string theory and it has been extensively studied both from physicists andmathematicians (see e.g. [1, 2, 6, 8, 12, 14, 18, 19, 20, 28, 43]).The Anomaly flow turned out to be a powerful tool in the study of the Hull-Strominger system.In particular, it was used to give an alternative proof of the outstanding results obtained by Fu andYau in [18, 19] about the existence of solutions to (2). More precisely, in [32] Phong, Picard andZhang studied the flow on a torus fibration over a K3 surface, showing that if ω is conformallybalanced and satisfies some extra assumptions, then the flow has a long-time solution which alwaysconverges to a solution of the Hull-Strominger system, once the connections ∇ τ and ∇ κ are bothChern.In a attempt to better understand the ‘general’ behaviour of the Anomaly flow, a simplifiedversion of the flow with ‘flat’ bundle was proposed in [31], namely ∂ t ( k Ψ k ω t ω t ) = i∂∂ω t − α ′ Rm τt ∧ Rm τt ) . (3) HE ANOMALY FLOW ON NILMANIFOLDS 3
Then, following the approach proposed by Fei and Yau in [11] to solve the Hull-Strominger systemon complex unimodular Lie groups, Phong, Picard and Zhang investigated this new flow on suchLie groups [34].In the present paper we study the behaviour of the Anomaly flows (1) and (3) on a class ofnilmanifolds. We will assume the trace
Tr( A κt ∧ A κt ) to be of a special type (see Assumption 3.1)and all the involved structures are to be intended invariant .Our first result characterizes the solutions to the Anomaly flows under our hypotheses. Theorem A.
Let M = Γ \ G be a -step nilmanifold of dimension with first Betti number b ≥ .Let J be a non-parallelizable complex structure on M . Then, there always exists a preferable (1 , -coframe { ζ i } on X = ( M, J ) such that the family of Hermitian metrics ω t solving (1) or (3) isgiven by ω t = i r ( t ) (cid:16) ζ + a ζ + b ζ + ¯ b ζ (cid:17) + i c ζ , for some a, c ∈ R and b ∈ C depending on ω . Furthermore, if ω t is a solution to the Anomalyflow (3) , then r ( t ) solves the ODE ddt r ( t ) = K + K r ( t ) , (4) with K , K ∈ R constants depending on K = K ( ω ) and K = K ( ω , α ′ , τ ) . As a direct consequence, we have that the qualitative behaviour of the Anomaly flow (3) onlydepends on signs of the constants K and K . Remarkably, this implies that the Anomaly flow (3)admits both immortal and ancient invariant solutions on the same nilmanifolds, and, as far as weknow, this provides the second example of a metric flow with such a property (the first one is thepluriclosed flow [3]). We also show that the constant K is closely related to a conformal invariantof the metric ω introduced and studied in [17].Our second result is about the convergence of the Anomaly flow (3) in Gromov-Hausdorff topol-ogy. In particular, under the same assumption of Theorem A, we have Theorem B.
Let ω t be an immortal solution to the Anomaly flow (3) . Then, ( X, (1 + t ) − ω t ) converges either to a point or to a real torus T in the Gromov-Hausdorff topology as t → + ∞ ,depending on the initial metric ω and the signs of K and K in (4) . It is worth noting that, Theorem A holds for any initial invariant Hermitian metric ω on X .Nonetheless, in view of the Hull-Strominger system, it would be desirable for the Anomaly flowsto preserve the locally conformally balanced condition. Under the assumptions of Theorem A, ourthird result states as follows Theorem C.
The Anomaly flows (1) and (3) preserve the balanced condition.
The last part of this paper is devoted to the study of the Anomaly flow (1) in some interestingnon-flat cases. In particular, we consider a class of Lie groups G arising from Theorem A, and weequip them with holomorphic tangent bundles, that is E = T , G . Theorem D.
If the initial metrics ( ω , H ) are both diagonal, then ω t and H t hold diagonal alongthe Anomaly flow (1) . HE ANOMALY FLOW ON NILMANIFOLDS 4
As a relevant application of this theorem, we obtain solutions to the field equations of the heteroticstring on the nilmanifold corresponding to the nilpotent Lie group N . In [23] Ivanov proved that,in order to solve such field equations, the solution to the Hull-Strominger system (2) must satisfythe extra condition that Rm is the curvature of an SU(3) -instanton with respect to ω (see also [13]).In our setting, this means that the curvature of the Gauduchon connection ∇ τ has to satisfy ω ∧ Rm τ = 0 , ( Rm τ ) , = ( Rm τ ) , = 0 . (5)Following [33], we will refer to the whole system given by (2) and (5) as the Hull-Strominger-Ivanovsystem . Explicit solutions to this system were found in [13] on nilmanifolds, and more recentlyin [28] on solvmanifolds and on the quotient of SL(2, C ).As an application of Theorem D, given the nilpotent Lie group N , we prove that if ∇ κt is theStrominger-Bismut connection of H t and the initial metric ω is balanced, then the flow reduces toan ODE of the form of (4). This allows us to prove that the Anomaly flow (1) always converges toa solution of the Hull-Strominger-Ivanov system when ∇ τt is the Strominger-Bismut connection ofthe metric ω t (see Theorem 5.8).It is worth noting that a generalization of the Anomaly flow to Hermitian manifolds of anydimension has been proposed in [35]; while, in [9] Fei and Phong proved that this generalizedAnomaly flow is related to a flow in the Hermitian curvature flows family, that is, a family ofparabolic flows introduced by Streets and Tian in [40]. We mention that the Hermitian curvatureflow related to the Anomaly flow has been studied on 2-step nilpotent complex Lie groups by thefirst named author in [37], who proved long-time existence and convergence results. We also refer to[3, 5, 25, 29, 36, 38, 45] for some recent results on Hermitian curvature flows in different homogeneoussettings.The paper is organized as follows. Section 2 is devoted to basic computations on our class ofnilpotent Lie groups. In particular, we show that under our assumptions we can always find apreferable real coframe on such Lie groups, namely an adapted basis . Then, by using this basis, weexplicitly compute Tr( Rm τ ∧ Rm τ ) . In Section 3 we begin the study of the Anomaly flows and weprove Theorem A and Theorem C. In Section 4 we focus on the Anomaly flow (3), showing that wecan always reduce the flow to the ODE (4). We also study the qualitative behaviour of the Anomalyflow (3) depending on the signs of K , K . These results will in turn imply Theorem B. In Section 5we study the Anomaly flow (1) on a special class of nilpotent Lie groups and we prove Theorem D.We also investigate the behaviour of the flow on an explicit example. Finally, Appendix A andAppendix B contain some technical computations which we used in the paper.2. Preliminaries
In this section, we consider a nilpotent Lie group G of (real) dimension 6 endowed with a left-invariant Hermitian structure ( J, ω ) . In particular, we will find a coframe which adapts to theHermitian structure, allowing us to explicitly compute the trace of the curvature for any Hermitianconnection in the Gauduchon family.Let G be a 6-dimensional Lie group equipped with a left-invariant complex structure J and aleft-invariant Hermitian metric ω . Let { Z , Z , Z } be a left-invariant (1,0)-frame of ( G, J ) , and { ζ , ζ , ζ } its dual frame. Then, we can always write ω = i ( r ζ + s ζ + k ζ ) + u ζ − ¯ u ζ + v ζ − ¯ v ζ + z ζ − ¯ z ζ , (6) HE ANOMALY FLOW ON NILMANIFOLDS 5 where r, s, k ∈ R ∗ and u, v, z ∈ C satisfy r s > | u | , s k > | v | , r k > | z | , (7)and i det ω = r s k + 2 Re ( i ¯ u ¯ vz ) − k | u | − r | v | − s | z | > (8)by the positive definiteness of the metric. Here det ω = 18 det i r u z − u i s v − z − v i k . In the following, we focus on Lie groups G which are 2-step nilpotent and such that ( G, J ) is nota complex Lie group , that is, the left-invariant complex structure J is not complex parallelizable.Notice that, the latter condition excludes just two cases (the complex torus and the Iwasawa mani-fold), both of which have already been studied in [34] (see also [37]). Additionally, we will supposethat the dimension of the first Chevalley-Eilenberg cohomology group H ( g ) of the Lie algebra g of G is at least 4. We denote such dimension by b ( g ) , since by the Nomizu theorem it coincideswith the first Betti number of the nilmanifold Γ \ G obtained as the quotient of G by a co-compactlattice Γ .Under these assumptions we are able to study a large family of Hermitian metrics in a unifiedsetting. Indeed, by means of [42, Proposition 2] (see also [4, Proposition 2.4]), if J is not complexparallelizable and b ( g ) = dim H ( g ) ≥ , then there exists a left-invariant (1,0)-coframe { ζ , ζ , ζ } on ( G, J ) satisfying ( dζ = dζ = 0 ,dζ = ρ ζ + ζ + λ ζ + ( x + i y ) ζ , (9)where x, y, λ ∈ R with λ ≥ , and ρ ∈ { , } . (See Table 1 in Section 4.2 for a description of all thereal Lie groups supporting such Hermitian structures.)2.1. Adapted bases.
Our first result shows that we can always find a preferable (real) left-invariant coframe { e , . . . , e } on G associated to any left-invariant Hermitian structure ( J, ω ) . In the following, we refer to sucha coframe as adapted basis . Proposition 2.1.
Let G be a -step nilpotent Lie group of dimension with b ( g ) ≥ , g beingthe Lie algebra of G . Let J be a left-invariant non-parallelizable complex structure on G and ω aleft-invariant J -Hermitian metric. Suppose that J and ω are defined by (9) and (6) , respectively.Then, there exists a (real) left-invariant coframe { e , . . . , e } on G , such that: (a) The complex structure J and the metric ω satisfy J e = − e , J e = − e , J e = − e , ω = e + e + e . (10) HE ANOMALY FLOW ON NILMANIFOLDS 6 (b)
The coframe satisfies the following structure equations de = de = de = de = 0 ,de = k e ∆ e ( ρ + λ ) e − k e ∆ e ( ρ − λ ) e + k e ∆ e (cid:0) r e y − λ u e (cid:1) e ,de = − k e r e e + k e u e r e ∆ e e + k e r e ∆ e (cid:0) r e ( ρ − λ ) + 2 u e (cid:1) e + k e r e ∆ e (cid:0) r e ( ρ + λ ) − u e (cid:1) e + k e u e r e ∆ e e , − k e r e ∆ e (cid:0) r e x − λ r e u e + u e + u e (cid:1) e . (11) Here, x, y, λ ∈ R with λ ≥ , and ρ ∈ { , } are the coefficients in (9) which define thecomplex structure J , whereas the coefficients r e , s e , k e , u e , u e ∈ R , which depends on thecoefficients of ω , are given by r e = r − | z | k , s e = s − | v | k , k e = k , u e + i u e := u e = u − i ¯ vzk . (12) The term ∆ e in the equations (11) stands for ∆ e := p r e s e − | u e | = q i det ωk . (c) The -form e is a positive multiple of the (2 , -form ζ , concretely e = 2 i det ωk ζ . (13) Proof.
Starting from a left-invariant (1 , -coframe { ζ , ζ , ζ } satisfying (9) and a generic J -Hermitian metric ω in the form of (6), we first consider the left-invariant (1,0)-coframe σ := ζ , σ := ζ , σ := ζ − ivk ζ − izk ζ . (14)This map defines an automorphism of the complex structure J which preserves the complex structureequations (9), i.e. the (1 , -coframe { σ , σ , σ } still satisfies dσ = dσ = 0 , dσ = ρ σ + σ + λ σ + ( x + i y ) σ . (15)With respect to this coframe, the Hermitian metric ω can be written as ω = i ( r σ σ + s σ σ + k σ σ ) + u σ σ − u σ σ , (16)where the metric coefficients are given by r σ = r − | z | k , s σ = s − | v | k , k σ = k , u σ = u − i ¯ vzk . (17)Notice that from (7) we have r σ , s σ , k σ > and r σ s σ > | u σ | .Let us now consider the left-invariant (1 , -coframe τ := r σ σ + i u σ r σ σ , τ := ∆ σ r σ σ , τ := k σ σ , (18)where ∆ σ := p r σ s σ − | u σ | . Then, a direct calculation yields that ω can be written as ω = i τ + i τ + i τ HE ANOMALY FLOW ON NILMANIFOLDS 7 and, by using (15), the complex structure equations become dτ = dτ = 0 ,dτ = ρ k σ ∆ σ τ + k σ r σ τ + k σ r σ ∆ σ (cid:0) iu σ + λr σ (cid:1) τ − ik σ u σ r σ ∆ σ τ + k σ r σ ∆ σ (cid:0) | u σ | − ir σ u σ λ + r σ x + i r σ y (cid:1) τ . (19)Finally, let us consider the real left-invariant coframe { e , . . . , e } on G given by e + i e := τ , e + i e := τ , e + i e := τ . (20)Then, with respect to this real coframe, (10) holds and hence (a) is proved.Now, let us set u σ + i u σ := u σ . By means of (19), a direct computation yields that the structureequations in terms of the real coframe { e , . . . , e } are given by de = de = de = de = 0 , ∆ σ k σ de = ( ρ + λ ) e − ( ρ − λ ) e + σ (cid:0) r σ y − λ u σ (cid:1) e , r σ ∆ σ k σ de = − σ e + 2 u σ e + (cid:0) r σ ( ρ − λ ) + 2 u σ (cid:1) e + (cid:0) r σ ( ρ + λ ) − u σ (cid:1) e +2 u σ e − σ (cid:0) r σ x − λr σ u σ + u σ + u σ (cid:1) e . Therefore, setting r e := r σ , s e := s σ , k e := k σ , u e := u σ (thus, u e := u σ and u e := u σ ) and ∆ e = ∆ σ , we get (11) and (b) follows. Notice that (12) is precisely (17) in the new notation.Moreover, from (12) we get ∆ e = r e s e − | u e | = 1 k (cid:0) r s k + 2 Re ( i ¯ u ¯ vz ) − k | u | − r | v | − s | z | (cid:1) = 8 i det ωk , due to (8).Finally, in order to prove (c), it is enough to note that e = τ = ∆ e ζ . (cid:3) Remark 2.2.
In the balanced case, adapted bases were found in [44, Theorem 2.11] by consideringa partition of the space of metrics into the subsets “ u = 0 ” and “ u = 0 ” for the given left-invariantmetrics (6). However, the study of the Anomaly flow requires to consider a global setting involvingthe whole space of left-invariant Hermitian metrics ω on ( G, J ) , as it has been obtained in ourProposition 2.1.2.2. Trace of the curvature.
In the following, we explicitly compute the trace of the curvature of a Hermitian connection in theGauduchon family {∇ τ } τ ∈ R for our class of nilpotent Lie groups. We will adopt the conventionused in [13] (see also [28]).Given a smooth manifold M equipped with a Hermitian structure ( J, ω ) , a connection ∇ on thetangent bundle T M is said to be a
Hermitian connection if ∇ J = 0 and ∇ ω = 0 . Gauduchonintroduced in [21] a 1-parameter family {∇ τ } τ ∈ R of canonical Hermitian connections, which can bedefined via ω ( J ( ∇ τX Y ) , Z ) = ω ( J ( ∇ LCX Y ) , Z ) + 1 − τ T ( X, Y, Z ) + 1 + τ C ( X, Y, Z ) , (21)where ∇ LC is the Levi-Civita connection of the Riemannian manifold ( M, ω ) , and T and C aregiven by T ( · , · , · ) := − dω ( J · , J · , J · ) and C ( · , · , · ) := dω ( J · , · , · ) . HE ANOMALY FLOW ON NILMANIFOLDS 8
These connections are distinguished by the properties of their torsion tensors and, in view of (21),the Chern connection ∇ c and the Strominger-Bismut connection ∇ + can be recovered by taking τ = 1 and τ = − , respectively.Now, let us consider a 6-dimensional Lie group G equipped with a left-invariant Hermitian struc-ture ( J, ω ) . Let also { e , . . . , e } be an adapted basis to the Hermitian structure on G , i.e. J and ω are expressed by (10). The connection -forms σ ij associated to any linear connection ∇ are σ ij ( e k ) := ω ( J ( ∇ e k e j ) , e i ) , that is, ∇ X e j = σ j ( X ) e + · · · + σ j ( X ) e ; while, the curvature -forms Ω ij of ∇ are given by Ω ij := dσ ij + X ≤ k ≤ σ ik ∧ σ kj . (22)Then, the trace of the 4-form Ω ∧ Ω can be defined via Tr(Ω ∧ Ω) = X ≤ i 12 ( ω ( J e i , [ e j , e k ]) − ω ( J e k , [ e i , e j ]) + ω ( J e j , [ e k , e i ])) = 12 ( c ijk − c kij + c jki ) , and hence, by means of (21), the connection 1-forms ( σ τ ) ij of ∇ τ are given by ( σ τ ) ij ( e k ) =( σ LC ) ij ( e k ) − − τ T ( e i , e j , e k ) − τ C ( e i , e j , e k )= 12 ( c ijk − c kij + c jki ) − − τ T ( e i , e j , e k ) − τ C ( e i , e j , e k )= 12 ( c ijk − c kij + c jki ) + 1 − τ dω ( J e i , J e j , J e k ) − τ dω ( J e i , e j , e k ) . Henceforth, the curvature of the Gauduchon connection ∇ τ will be denoted indistinctly by Rm τ or Ω τ . We are now in a position to compute the trace of Ω τ ∧ Ω τ for our class of nilpotent Liegroups. In our next proposition, we show that the trace is of a special type. This will allow us tosubstantially simplify the Anomaly flow equations. Proposition 2.3. Let G be a -step nilpotent Lie group of dimension with b ( g ) ≥ , g beingthe Lie algebra of G . Let J be a left-invariant non-parallelizable complex structure on G and ω aleft-invariant J -Hermitian metric. Suppose that J and ω are defined, respectively, by (9) and (6) interms of a left-invariant (1 , -coframe { ζ l } l =1 . Then, for any Gauduchon connection ∇ τ , the traceof its curvature satisfies Tr(Ω τ ∧ Ω τ ) = C ζ , HE ANOMALY FLOW ON NILMANIFOLDS 9 where C = C ( ρ, λ, x, y ; r, s, k, u, v, z ; τ ) is a constant depending both on the Hermitian structure andthe connection. More precisely, we have Tr(Ω τ ∧ Ω τ ) = − ( τ − k e r e s e − | u e | ) nh ( ρ − λ + 5 x )( s e − λs e u e + 2 x | u e | ) − λ x ( u e − u e ) − λu e y ( s e − λu e ) + 6 y | u e | + τ ( ρ + λ − x )( s e − λs e u e + 2 x | u e | )+ τ (cid:16) ( − ρ + x )( s e − λs e u e + 2 x | u e | ) − λ x ( u e − u e ) − λu e y ( s e − λu e ) + 2 y | u e | (cid:17)i + r e λ h ( ρ − λ + 2 x )( λs e − u e x − u e y ) − u e ( x + y )+ τ ( ρ + λ − x )( λs e − u e x − u e y )+ τ (cid:16) − ρ ( λs e − u e x − u e y ) − u e ( x + y ) (cid:17)i + r e ( x + y ) h ( ρ − λ + 5 x ) + τ ( ρ + λ − x ) + τ ( − ρ + x ) i o ζ , where r e , s e , k e and u e = u e + i u e are given in Proposition 2.1.Proof. By means of Proposition 2.1, there always exists an adapted basis { e , . . . , e } on the Liegroup G for the left-invariant Hermitian structure ( J, ω ) . Therefore, let ( σ τ ) ij be the connection1-forms of the Gauduchon connection ∇ τ in this basis. Since ∇ τ is Hermitian, then the forms ( σ τ ) ij satisfy the condition ( σ τ ) ij = − ( σ τ ) ji . Moreover, a direct computation yields the followingconnection 1-forms: ( σ τ ) = − k e r e ( τ − e , ( σ τ ) = λ k e √ r e s e −| u e | ( τ − e + k e u e r e √ r e s e −| u e | ( τ − e , ( σ τ ) = − k e ( λ r e − u e )2 r e √ r e s e −| u e | ( τ − e , ( σ τ ) = − k e r e ( τ +1) e + k e r e √ r e s e −| u e | (cid:0) ρr e ( τ − u e − λr e )( τ +1) (cid:1) e − k e u e r e √ r e s e −| u e | ( τ +1) e , ( σ τ ) = k e r e ( τ +1) e − k e u e r e √ r e s e −| u e | ( τ +1) e + k e r e √ r e s e −| u e | (cid:0) ρr e ( τ − − ( u e − λr e )( τ +1) (cid:1) e , ( σ τ ) = − k e ( λ u e − r e y ) r e s e −| u e | ( τ − e − k e ( | u e | − λ r e u e + r e x ) r e ( r e s e −| u e | ) ( τ − e , ( σ τ ) = − k e ( ρ r e ( τ − − u e ( τ +1))2 r e √ r e s e −| u e | e + k e u e ( τ +1)2 r e √ r e s e −| u e | e − k e ( | u e | − λ r e u e + r e x )2 r e ( r e s e −| u e | ) ( τ +1) e + k e ( λ u e − r e y )2( r e s e −| u e | ) ( τ +1) e , ( σ τ ) = k e u e ( τ +1)2 r e √ r e s e −| u e | e − k e ( ρ r e ( τ − u e ( τ +1))2 r e √ r e s e −| u e | e + k e ( λ u e − r e y )2( r e s e −| u e | ) ( τ +1) e + k e ( | u e | − λ r e u e + r e x )2 r e ( r e s e −| u e | ) ( τ +1) e , ( σ τ ) =0 , HE ANOMALY FLOW ON NILMANIFOLDS 10 together with the following relations: ( σ τ ) = − ( σ τ ) , ( σ τ ) = ( σ τ ) , ( σ τ ) = − ( σ τ ) , ( σ τ ) = ( σ τ ) , ( σ τ ) = − ( σ τ ) , ( σ τ ) = ( σ τ ) . Finally, by using the curvature 2-forms (Ω τ ) ij explicitly given in Appendix A and equation (23), theresult follows from a long but direct calculation. (cid:3) Let G be a nilpotent Lie group endowed with a left-invariant complex structure J defined by (9).Moreover, let us consider the following left-invariant closed (3,0)-form Ψ := ζ ∧ ζ ∧ ζ . (24)Given the Chern connection ∇ c of the metric ω , i.e. τ = 1 in the Gauduchon family, one alwayshas ∇ c Ψ = 0 ; whereas, by the proof of the previous proposition, one gets Corollary 2.4. Under the hypotheses of Proposition 2.3. If τ = 1 , then ∇ τ Ψ = 0 if and only if theHermitian metric ω is balanced.Proof. In terms of an adapted basis { e l } l =1 as in Proposition 2.1, we have that ∇ τ (( e + i e ) ∧ ( e + i e ) ∧ ( e + i e )) = 0 if and only if the connection 1-forms satisfy ( σ τ ) + ( σ τ ) + ( σ τ ) = 0 . Since the left-invariant(3,0)-forms are related by Ψ = c ( e + i e ) ∧ ( e + i e ) ∧ ( e + i e ) for some non-zero constant c ,we get that ∇ τ Ψ = 0 if and only if ( σ τ ) + ( σ τ ) + ( σ τ ) = − k e ( τ − r e s e − | u e | (cid:16) ( λ u e − r e y ) e + ( s e − λ u e + r e x ) e (cid:17) = 0 , where we have made use of the connection 1-forms given in the proof of Proposition 2.3. Therefore,given τ = 1 , the above equality holds if and only if λ u e − r e y = 0 = s e − λ u e + r e x. (25)On the other hand, by means of the structure equations in the adapted basis, one directly getsthat the metric ω satisfies the balanced condition dω = 0 if and only if ω ∧ dω = ( e + e ) ∧ de , which is equivalent to (25). (cid:3) The first Anomaly flow equation on nilpotent Lie groups We now study the behaviour of a general solution to the first equation in the Anomaly flow for ourclass of nilpotent Lie groups, under certain assumptions. In particular, since we focus on invariantsolutions, the first statement in Theorem A and Theorem C will follow, respectively, by Theorem3.7 and Theorem 3.4 below.Let G be a 6-dimensional -step nilpotent Lie group with b ≥ , endowed with a left-invariantnon-parallelizable complex structure J . Let { ζ , ζ , ζ } be a left-invariant (1 , -coframe on ( G, J ) satisfying (9) and let Ψ := ζ ∧ ζ ∧ ζ be the left-invariant closed (3,0)-form defined in (24). HE ANOMALY FLOW ON NILMANIFOLDS 11 Assumption 3.1. Let ( ω t , H t ) be the couple of left-invariant Hermitian metrics solving the Anomalyflow (1) and let Tr( A t ∧ A t ) be a multiple of the (2,2)-form ζ . Remark 3.2. Since we are considering left-invariant metrics, the evolution equation of ω t reducesto an ODE on the Lie algebra level. We stress that, all the stated results will also hold for theAnomaly flow (3) with flat bundle.Let ω t be a solution to the Anomaly flow (1) on ( G, J ) given by ω t = i (cid:16) r ( t ) ζ + s ( t ) ζ + k ( t ) ζ (cid:17) + 12 u ( t ) ζ − u ( t ) ζ + 12 v ( t ) ζ − v ( t ) ζ + 12 z ( t ) ζ − z ( t ) ζ , (26)where r ( t ) , s ( t ) , k ( t ) are positive real functions and u ( t ) , v ( t ) , z ( t ) are complex functions. Then, wehave Proposition 3.3. Let Assumption 3.1 hold. Then, for every Gauduchon connection ∇ τt on ( G, ω t ) ,the evolution equation of ω t in the Anomaly flow reduces to the ODE ddt ( k Ψ k ω t ω t ) = K ( t, α ′ , τ ) ζ , (27) with K ( t, α ′ , τ ) depending on the structure equations (9) of ( G, J ) and on the curvature A t of theconnection on ( E, H t ) .Proof. By means of (9) and (26), a direct computation yields that i∂ ¯ ∂ω t = − k ( t ) ( ¯ ∂ζ ∧ ∂ζ ¯3 − ∂ζ ∧ ¯ ∂ζ ¯3 ) = k ( t ) ( ρ + λ − x ) ζ . (28)On the other hand, by means of Proposition 2.3, the trace of the curvature of a Gauduchon con-nection ∇ τ on ( G, ω t ) satisfies Tr( Rm τ ∧ Rm τ ) = C ( t ) ζ . Therefore, by the assumption on the curvature A t , we have ∂ t ( k Ψ k ω t ω t ) = i∂∂ω t − α ′ Rm τ ∧ Rm τ ) − Tr( A t ∧ A t )) = K ( t, α ′ , τ ) ζ , where K ( t, α ′ , τ ) also depends on ( G, J ) and A t . (cid:3) Let us now analyze equation (27) in more detail. A direct computation yields that ω t ∧ ω t = (cid:0) r ( t ) s ( t ) −| u ( t ) | (cid:1) ζ + (cid:0) r ( t ) k ( t ) −| z ( t ) | (cid:1) ζ + (cid:0) s ( t ) k ( t ) −| v ( t ) | (cid:1) ζ − i (cid:16) r ( t ) v ( t ) − i z ( t ) u ( t ) (cid:17) ζ + i (cid:16) r ( t ) v ( t ) + i u ( t ) z ( t ) (cid:17) ζ + i (cid:0) s ( t ) z ( t ) + i u ( t ) v ( t ) (cid:1) ζ − i (cid:16) s ( t ) z ( t ) − i u ( t ) v ( t ) (cid:17) ζ − i (cid:16) k ( t ) u ( t ) − i z ( t ) v ( t ) (cid:17) ζ + i (cid:16) k ( t ) u ( t ) + i v ( t ) z ( t ) (cid:17) ζ . Therefore, by substituting in (27), one gets that the following relations hold along the flow: ddt (cid:0) k Ψ k ω t ( r ( t ) s ( t ) −| u ( t ) | ) (cid:1) = 2 K ( t, α ′ , τ ) , (29)and ddt (cid:0) k Ψ k ω t ( r ( t ) k ( t ) −| z ( t ) | ) (cid:1) = 0 = ⇒ r ( t ) k ( t ) −| z ( t ) | = c k Ψ k ω t , (30) HE ANOMALY FLOW ON NILMANIFOLDS 12 ddt (cid:0) k Ψ k ω t ( s ( t ) k ( t ) −| v ( t ) | ) (cid:1) = 0 = ⇒ s ( t ) k ( t ) −| v ( t ) | = c k Ψ k ω t , (31) ddt (cid:16) k Ψ k ω t ( r ( t ) v ( t ) − i z ( t ) u ( t )) (cid:17) = 0 = ⇒ r ( t ) v ( t ) − i z ( t ) u ( t ) = c k Ψ k ω t , (32) ddt (cid:0) k Ψ k ω t ( s ( t ) z ( t ) + i u ( t ) v ( t )) (cid:1) = 0 = ⇒ s ( t ) z ( t ) + i u ( t ) v ( t ) = c k Ψ k ω t , (33) ddt (cid:16) k Ψ k ω t ( k ( t ) u ( t ) − i z ( t ) v ( t )) (cid:17) = 0 = ⇒ k ( t ) u ( t ) − i z ( t ) v ( t ) = c k Ψ k ω t , (34)for some constants c , c ∈ R with c , c > , and c , c , c ∈ C , which are determined by the initialmetric ω .3.1. Special Hermitian metrics along the flow. In [30] Phong, Picard and Zhang proved that the Anomaly flow preserve the conformally balancedcondition, once the connections ∇ τ and ∇ κ are both Chern. In the following, we extend such aresult to any connection in the Gauduchon family for our class of nilpotent Lie groups. Moreover,we also show that the locally conformally Kähler condition is preserved along the flow.A Hermitian metric ω is said to be locally conformally Kähler if it is conformal to some localKähler metric in a neighborhood of each point. Recall that such metrics are also characterized bythe condition dω = θ ∧ ω , where θ is the (closed) Lee form. Theorem 3.4. Under Assumption 3.1, we have: (i) If ω is balanced, then ω t remains balanced along the Anomaly flow. (ii) If ω is locally conformally Kähler, then ω t remains locally conformally Kähler along theAnomaly flow. Remark 3.5. By [15, Theorem 1.2], the pluriclosed condition ∂ ¯ ∂ω = 0 for a left-invariant metric ω on ( G, J ) only depends on the complex structure J (see also [42]). Therefore, if the initial metric ω is pluriclosed, the solution ω t to the Anomaly flow holds pluriclosed. Proof of Theorem 3.4. In view of [42, Proposition 25], the left-invariant Hermitian metric ω t isbalanced if and only if s ( t ) k ( t ) − | v ( t ) | + ( x + i y ) (cid:0) r ( t ) k ( t ) − | z ( t ) | (cid:1) = i λ (cid:16) k ( t ) u ( t ) + i v ( t ) z ( t ) (cid:17) . (35)On the other hand, by means of (30) and (31), the left-hand side of (35) reduces to s ( t ) k ( t ) − | v ( t ) | + ( x + i y ) (cid:0) r ( t ) k ( t ) − | z ( t ) | (cid:1) = c ( x + i y ) + c k Ψ k ω t , while, by means of (34), the right-hand side of (35) is equal to i λ (cid:16) k ( t ) u ( t ) + i v ( t ) z ( t ) (cid:17) = i λ ¯ c k Ψ k ω t . Thus, ω t is a balanced metric if and only if c ( x + i y ) + c = i λ ¯ c . Since the constants c , c and c only depend on the initial metric ω , it follows that ω t satisfies the balanced condition if and onlyif ω does.Let us now prove (ii). By means of [42, Proposition 32], if ω is locally conformally Kähler, wemust have ρ = λ = y = 0 and x = 1 HE ANOMALY FLOW ON NILMANIFOLDS 13 in the complex structure equations (9). Moreover, ω t is locally conformally Kähler if and only if r ( t ) k ( t ) −| z ( t ) | = s ( t ) k ( t ) −| v ( t ) | and k ( t ) u ( t ) = i z ( t ) v ( t ) . Therefore, in view of (30), (31) and (34), it follows that ω t is a locally conformally Kähler metric ifand only if c − c = c = 0 . Finally, since the constants c , c , c only depend on the initial metric ω , we get that ω t is locally conformally Kähler if and only if ω is locally conformally Kähler aswell, and the claim follows. (cid:3) Reduction to almost diagonal initial metrics and the general solution. In the following, we prove that any initial Hermitian metric ω can be taken to be almost diagonal.Then, we use this result to obtain the general solution ω t to the first evolution equation in theAnomaly flow (1).A Hermitian metric ω is said to be almost diagonal if its metric coefficients satisfy v = z = 0 in (6). Our next result shows that we can always choose a preferable (1 , -coframe on ( G, J ) suchthat the metric ω is almost diagonal. Lemma 3.6. Let ω be a left-invariant Hermitian metric on ( G, J ) . Then, there exists an automor-phism which preserves both the complex structure equations (9) and the (3,0)-form Ψ , and such that ω reduces to an almost diagonal form.Proof. To prove this lemma we essentially use the same argument as in Proposition 2.1. Let usconsider the automorphisms induced by (14). By construction this automorphism preserves thecomplex structure equations (see (15)). Moreover, a direct computation yields that Ψ = ζ ∧ ζ ∧ ζ = σ ∧ σ ∧ σ and hence the holomorphic (3,0)-form Ψ is also preserved. Finally, in terms of the coframe { σ l } l =1 the Hermitian metric ω expresses as in (16) and hence the claim follows. (cid:3) We are now in a position to describe the general solution to the Anomaly flow starting from analmost diagonal metric. Theorem 3.7. Under Assumption 3.1, the Anomaly flow preserves the almost diagonal condition.More concretely, if ω is almost diagonal, then ω t evolves as ω t = i (cid:18) r ( t ) ζ + c c r ( t ) ζ + c c −| c | ζ (cid:19) + c c r ( t ) ζ − ¯ c c r ( t ) ζ , (36) where c , c > and c ∈ C with c c > | c | , are constants determined by the initial metric ω .Furthermore, k Ψ k ω t = 8 c ( c c −| c | ) r ( t ) . Proof. Since equations (32) and (33) hold, the functions v ( t ) and z ( t ) satisfy v ( t ) = c s ( t ) + i c u ( t ) k Ψ k ω t ( r ( t ) s ( t ) − | u ( t ) | ) , z ( t ) = − i c u ( t ) + c r ( t ) k Ψ k ω t ( r ( t ) s ( t ) − | u ( t ) | ) , for any t in the defining interval. On the other hand, since we assumed ω to be almost diagonal,i.e. v (0) = z (0) = 0 , we get c = c = 0 . Therefore, v ( t ) = 0 and z ( t ) = 0 and hence the solution ω t holds almost diagonal. HE ANOMALY FLOW ON NILMANIFOLDS 14 Let us now prove the second part of the statement. As a direct consequence of (30), (31) and(34), it follows that ( r ( t ) s ( t ) − | u ( t ) | ) k ( t ) = c c − | c | k Ψ k ω t , which implies k Ψ k ω t = c c − | c | ( r ( t ) s ( t ) − | u ( t ) | ) k ( t ) , with c c − | c | > by the positive definiteness of the metric ω . Moreover, by the definition of k Ψ k ω t , we have k Ψ k ω t = 1det ω t = 8( r ( t ) s ( t ) − | u ( t ) | ) k ( t ) and hence k ( t ) = r c c − | c | . In particular, k ( t ) is constant. Finally, by means of (30), (31) and (34), we have c r ( t ) k ( t ) − c s ( t ) k ( t ) = ( c r ( t ) − c s ( t ) ) k ( t ) and c r ( t ) k ( t ) − c u ( t ) k ( t ) = ( c r ( t ) − c u ( t )) k ( t ) , which respectively imply s ( t ) = c c r ( t ) and u ( t ) = c c r ( t ) , and the claim follows. (cid:3) When the initial metric ω is diagonal (that is, u (0) = v (0) = z (0) = 0 ), the above result simplifies to Corollary 3.8. Under Assumption 3.1, the Anomaly flow preserves the diagonal condition. Specif-ically, if ω is diagonal, then ω t is given by ω t = i (cid:18) r ( t ) ζ + c c r ( t ) ζ + c c ζ (cid:19) , where c = √ r (0) k (0) s (0) > and c = √ s (0) k (0) r (0) > . Moreover, k Ψ k ω t = c r ( t ) . Our next result describes the evolution of the trace Tr( Rm τt ∧ Rm τt ) along the Anomaly flow,under the assumption for the initial metric ω to be almost diagonal. Proposition 3.9. Under the hypotheses of Theorem 3.7, the trace of the curvature of the Gauduchonconnection ∇ τ of ( G, ω t ) satisfies Tr( Rm τt ∧ Rm τt ) = Cr ( t ) ζ , where C = C ( ω , τ ) is a constant depending only on the initial metric ω and the connection ∇ τ .Proof. The result is a consequence of Proposition 2.3. Indeed, the coefficients r e , s e , k e and u e = u e + i u e appearing in Proposition 2.3 are related to the coefficients of the metric ω t via (12).On the other hand, by means of Theorem 3.7, the metric ω t holds almost diagonal and by (12) and(36) we get r e = r ( t ) , s e = c c r ( t ) , k e = c c − | c | , u e = c c r ( t ) . (37)Therefore, the claim follows by (37) and the formula of the trace given in Proposition 2.3. (cid:3) HE ANOMALY FLOW ON NILMANIFOLDS 15 The Anomaly flow with flat holomorphic bundle We now focus on the Anomaly flow (3). In particular, we show that this flow always reduces to asingle ODE, which we call model problem , and the second part of Theorem A will directly follow byTheorem 4.1 below. Moreover, the qualitative behaviour of the model problem will be investigated.Let G be a 6-dimensional -step nilpotent Lie group with b ≥ , equipped with a left-invariantnon-parallelizable complex structure J . Let { ζ , ζ , ζ } be a left-invariant (1 , -coframe on ( G, J ) satisfying (9) and let Ψ be the left-invariant closed (3,0)-form defined in (24), i.e. Ψ := ζ ∧ ζ ∧ ζ .In view of Proposition 3.3 and Theorem 3.7, the coefficient r ( t ) of the metric ω t in (36) evolvesas ∂ t r ( t ) = c K ( t, α ′ , τ ) , where the right-hand side is given by K ( t, α ′ , τ ) ζ = i∂∂ω t − α ′ τt ∧ Ω τt ) . On the other hand, by means of (28) and Theorem 3.7, we get i∂∂ω t = B ζ , for the constant B = B ( ω ) = c c −| c | ( ρ + λ − x ) ∈ R ; while, by means of Proposition 3.9, wehave Tr(Ω τt ∧ Ω τt ) = Cr ( t ) ζ , for a constant C = C ( ω , τ ) ∈ R . Therefore, we get c K ( t, α ′ , τ ) = K + K r ( t ) , where K = c B and K = − α ′ c C , and the following theorem holds. Theorem 4.1. The Anomaly flow (3) is equivalent to the model problem ddt r ( t ) = K + K r ( t ) , (38) where K , K ∈ R are constants depending on K = K ( ω ) and K = K ( ω , α ′ , τ ) . Qualitative behaviour of the model problem. We now investigate the qualitative behaviour of the model problem (38), which can be rewritten as h ′ ( t ) = K + K h ( t ) , h ( t ) > . (39)A solution h ( t ) to (39) is said to be immortal , eternal or ancient if the defining interval ( T − , T + ) isequal to ( − ε, + ∞ ) , ( −∞ , + ∞ ) or ( −∞ , ε ) for some ε > , respectively.When either K = 0 or K = 0 the ODE (39) can be explicitly solved, otherwise we work asfollows. HE ANOMALY FLOW ON NILMANIFOLDS 16 • K > and K > . Under these assumptions, (39) does not admit any stationary point. Nonetheless, we have thefollowing Proposition 4.2. Any solution h ( t ) to the model problem (39) is immortal. In particular, h ( t ) ∼ K · t as t → + ∞ .Proof. Let h ( t ) be a solution to the model problem (39). Since h ′ ( t ) = K + K h ( t ) > , it follows that h ( t ) ≥ h (0) , for every t ∈ [0 , T + ) . On the other hand, h ′ ( t ) ≤ K + K h (0) and the long-time existence follows, since h ( t ) ≤ c t + h (0) with c := K + K h (0) .Let us now suppose by contradiction that h ′ ( t ) → as t → + ∞ . Then, this would imply lim t →∞ (cid:18) K + K h ( t ) (cid:19) = 0 , which is not possible since K , K > . Therefore, we have lim t →∞ h ′ ( t ) = K and hence h ( t ) ∼ K · t as t → + ∞ . Finally, a similar argument shows that if the solution existsbackward in time for any t < , then h ( t ) ∼ K · t , as t → −∞ , which is not possible since h ( t ) > . (cid:3) • K > and K < . Let us denote by h := p − K /K . Then, we have Proposition 4.3. Let h ( t ) be a solution to the model problem (39) . It follows that (i) if h (0) = h , then the solution is stationary; (ii) if h (0) > h , then the solution is eternal and h ( t ) ∼ K · t as t → + ∞ ; (iii) if h (0) < h , then the solution is ancient.Furthermore, any solution detects the stationary point as t → −∞ .Proof. Let h ( t ) be the solution to (39). Then, a direct computation yields that h is the uniquestationary point to the flow, and the first claim follows.Now, let us suppose h (0) > h . Then, there exists ε > such that h (0) = q − K K + ε . Therefore,we have h ′ (0) = εK − K + εK > , and hence h ( t ) ′ > for every t ∈ ( T − , T + ) . On the other hand, h ′ ( t ) ≤ K = ⇒ h ( t ) ≤ K t + h (0) , for any t ≥ , HE ANOMALY FLOW ON NILMANIFOLDS 17 and the long-time existence follows. Moreover, since h ( t ) is always increasing and h is the uniquestationary point to the flow, it follows h ( t ) → h as t → −∞ . Thus, the solution h ( t ) is eternal.Finally, let us assume by contradiction that h ′ ( t ) → as t → + ∞ . Then, this would be equivalentto require lim t →∞ K + K h ( t ) = 0 , which is not possible since h (0) > h , and hence lim t →∞ h ′ ( t ) = K proves the second claim.Now, let us assume h (0) = q − K K − ε < h for some ε > . Then, a direct computation yieldsthat h ′ (0) = − εK − K + εK < , which in turn implies h ′ ( t ) < for every t ∈ ( T − , T + ) . On the other hand, it follows h ( t ) ≤ − εK − K + εK t + h (0) , for any t ≥ , and hence T + < + ∞ . Moreover, since h ( t ) is decreasing, we have lim t → T + h ′ ( t ) = lim t → T + (cid:18) K + K h ( t ) (cid:19) = −∞ . Finally, since h ( t ) is always decreasing and there exists a unique stationary solution h to the flow,we have that h ( t ) → h as t → −∞ and the last claim follows. (cid:3) • K < and K < . Under these assumptions, we have Proposition 4.4. Any solution h ( t ) to (39) is ancient. In particular, h ( t ) ∼ − K · t as t → −∞ . The proof of this result can be easily recovered using the same arguments as in Proposition 4.2. • K < and K > . Arguing in the same way of Proposition 4.3, we get Proposition 4.5. Let h ( t ) be a solution to (39) . It follows that (i) if h (0) = h , then the solution is stationary; (ii) if h (0) > h , then the solution is eternal and h ( t ) ∼ − K · t as t → −∞ ; (iii) if h (0) < h , then the solution is immortal.Furthermore, any solution detects the stationary point as t → + ∞ . HE ANOMALY FLOW ON NILMANIFOLDS 18 The sign of K and its relation to the Fu-Wang-Wu conformal invariant. We now investigate the relation between the constant K appearing in the model problem (38) andthe conformal invariant of ω introduced and studied by Fu, Wang and Wu in [17]. We also studythe sign of K in our class of nilpotent Lie groups.Let X be a compact n -dimensional complex manifold and ω a Hermitian metric on X . In [17],the notion of Gauduchon metric has been generalized by the so-called k -th Gauduchon equation ∂∂ω k ∧ ω n − k − = 0 , ≤ k ≤ n − . Then, since the k -th Gauduchon equation may not admit a solution, Fu, Wang and Wu consideredthe equation (in the conformal class of ω ) given by i ∂∂ ( e v ω k ) ∧ ω n − k − = γ k ( ω ) e v ω n , ≤ k ≤ n − , (40)proving that there always exist a unique constant γ k ( ω ) and a function v ∈ C ∞ ( X ) (unique up toa constant) satisfying (40). Moreover, the constant γ k ( ω ) is invariant under biholomorphisms andit smoothly depends on the metric ω , and its sign is invariant in the conformal class of ω [17].Now, let ( X, ω ) be a compact non-Kähler Hermitian manifold. In view of [24, Lemma 3.7] and[24, Proposition 3.8], for any ≤ k ≤ n − it follows that(i) if ω is balanced, then the constant γ k ( ω ) > ;(ii) if ω is locally conformally Kähler, then the constant γ k ( ω ) < .Therefore, we can apply these results to compute the sign of K = K ( ω ) in the model problem (38).Indeed, by [26, Proposition 2.7], any left-invariant Hermitian metric ω on ( G, J ) given by (6)satisfies i ∂ ¯ ∂ω ∧ ω = k i det ω (cid:0) ρ + λ − x (cid:1) ω , and hence γ ( ω ) = k i det ω (cid:0) ρ + λ − x (cid:1) . On the other hand, since K ( ω ) = c k ( ρ + λ − x ) in the model problem (38), we get K ( ω ) = c i det ω k γ ( ω ) . In particular, the sign of K ( ω ) is equal to the one of γ ( ω ) , which is an invariant of the conformalclass of ω . Actually, in our context we have that sign K = sign ( ρ + λ − x ) , and hence it only depends on the complex structure J . This fact for γ was first noticed in [16].Moreover, since an invariant Hermitian metric on a complex nilmanifold of complex dimension 3 is -st Gauduchon if and only if it is pluriclosed [16, Proposition 3.3], we have the following proposition. Proposition 4.6. The sign of K in the model problem (38) only depends on the complex structure J on G . Moreover: (i) If ω is balanced, then K > . (ii) If ω is locally conformally Kähler, then K < . (iii) The metric ω is pluriclosed if and only if K = 0 . HE ANOMALY FLOW ON NILMANIFOLDS 19 In Table 1, we provide the classification of the Lie groups admitting a complex structure satisfy-ing (9), together with the sign of the invariant K .The first column of Table 1 describes the nilpotent Lie algebra associated to the Lie group. Herewe use the notation for which the algebras are named as n k and then described (see e.g. [42]).Moreover, we denote by N k the Lie group corresponding to n k (second column of the table). Aboutthe other columns, we use the following convention. The symbol “ X ” means that the sign of K is theone given in the table for any complex structure on the corresponding Lie group N k , whereas “ X ( J ) ”means that there exist complex structures J on N k such that the sign of K is the one described bythe column. Therefore, different complex structures may lead to different sings on the same group N k . Finally, we use “ − ” to denote that there are no complex structures of the given sign.Lie algebra Lie group K < K = 0 K > n = (0 , , , , , N X ( J ) X ( J ) X ( J ) n = (0 , , , , , N X ( J ) − X ( J ) n = (0 , , , , , N X ( J ) X ( J ) X ( J ) n = (0 , , , , , N X ( J ) X ( J ) X ( J ) n = (0 , , , , , N − − X n = (0 , , , , , N − X − Table 1. The sign of K Let us recall that the groups N , N , N , N and N admit left-invariant balanced metrics, while N is the unique group admitting locally conformally Kähler metrics [42]. We refer to [26] for aclassification of the complex structures satisfying K < , = 0 or > . Remark 4.7. The nilpotent Lie group N is given by the product of R with the 5-dimensionalgeneralized Heisenberg group, while N is the real Lie group underlying the Iwasawa manifold.We stress that, by an appropriate choice either of the Gauduchon connection ∇ τ or of the slopeparameter α ′ , the sign of the constant K = K ( ω , α ′ , τ ) in the model problem (38) can take anyvalue. Therefore, the results presented in Section 4.1 apply to every nilpotent Lie group in Table 1.In particular, we get Proposition 4.8. Any Lie group in Table 1 with K = 0 admits both immortal and ancient left-invariant solutions to the Anomaly flow (3) . This result also extends to nilmanifolds arising from the quotient of a Lie group N k by a co-compact lattice.4.3. Convergence of the nilmanifolds. We are now in a position to prove our convergence result. Note that, a main ingredient in theproof of Theorem B will be given by the qualitative behaviour of the model problem studied inSection 4.1, together with Theorem 3.7.Let us recall that a family of compact metric spaces ( X t , d t ) converges to a metric space ( ¯ X, ¯ d ) in Gromov-Hausdorff topology as t → T , if for any increasing sequence t n → T there exists a HE ANOMALY FLOW ON NILMANIFOLDS 20 sequence of ε t n -approximations ϕ t n : X t n → ¯ X satisfying ε t n → . By definition, ϕ : X → ¯ X is an ε - approximation if | d t ( x, x ′ ) − ¯ d ( ϕ ( x ) , ϕ ( x ′ )) | < ε , for any x, x ′ ∈ X , and for all y ∈ ¯ X there exists x ∈ X such that ¯ d ( y, ϕ ( x )) < ε (see e.g. [39]). Proof of Theorem B. Let M = Γ \ G be a nilmanifold arising from our class of nilpotent Lie groupsand let { ζ , ζ , ζ } be an invariant (1,0)-frame of X = ( M, J ) . By means of (9), M gives rise to afibration over a real 4-dimensional tours π : M → T with fibers spanned by the real and imaginarypart of ζ . On the other hand, by means of Theorem 3.7 and the results presented in Section 4.1,one gets that either { ζ , ζ , ζ } or { ζ } shrink to zero along (1 + t ) − ω t as t → ∞ , depending onthe signs of K and K , and hence the claim follows. (cid:3) Evolution of the holomorphic vector bundle In this section we study the Anomaly flow (1) on a class of Lie groups belonging to (9). Explicitcomputations will be performed on the nilpotent Lie group N . In particular, we will prove thatunder certain choices of initial metric and connections, the Anomaly flow converges to a (non-flat)solution of the Hull-Strominger-Ivanov system.Let G be a -dimensional Lie group and let J be a left-invariant non-parallelizable complexstructure on G . Let us suppose that there exists a left-invariant (1 , -coframe { ζ , ζ , ζ } on G satisfying the structure equations ( dζ = dζ = 0 ,dζ = ρ ζ + ζ + ( x + i y ) ζ , (41)where x, y ∈ R and ρ ∈ { , } (i.e. we are considering λ = 0 in (9)). Let also the holomorphicvector bundle be E := T , G , and Ψ := ζ ∧ ζ ∧ ζ . Moreover, let the left-invariant Hermitian metrics ( ω , H ) be both diagonal, i.e. ω = i (cid:16) r ζ + s ζ + k ζ (cid:17) and H = i (cid:16) ˜ r ζ + ˜ s ζ + ˜ k ζ (cid:17) . Then, our main result is the following Theorem 5.1. The left-invariant metrics ω t and H t solving the Anomaly flow (1) remain diagonalalong the flow, and the coefficients of H t evolve via HE ANOMALY FLOW ON NILMANIFOLDS 21 ddt ˜ r ( t ) = 13 c c h (cid:0) c ( κ + 1) − c ρ ( κ − (cid:1) r ( t ) ˜ k ( t ) − c c ( κ − c x + c )˜ r ( t ) i ˜ k ( t ) r ( t ) ˜ r ( t ) ,ddt ˜ s ( t ) = 13 c c h (cid:0) c ( κ + 1) ( x + y ) − c ρ ( κ − (cid:1) r ( t ) ˜ k ( t ) − c ( κ − (cid:0) c ( x + y ) + c x (cid:1) ˜ s ( t ) i ˜ k ( t ) r ( t ) ˜ s ( t ) ,ddt ˜ k ( t ) = 23 c c h ρ ( κ − (cid:0) c ˜ s ( t ) + c ˜ r ( t ) (cid:1) − c c ( κ + 1) (cid:0) ( x + y )˜ r ( t ) + ˜ s ( t ) (cid:1)i ˜ k ( t ) r ( t ) ˜ r ( t ) ˜ s ( t ) . (42)To prove our statement, we need the following lemma. Lemma 5.2. Under the hypotheses of Theorem 5.1, Tr( A κ ∧ A κ ) = C ζ , where C = C ( λ, x, y ; ω , H ; κ ) is a constant depending both on the Hermitian structures and theconnection ∇ κ .Proof. The proof directly follows by Lemma 7.1 in Appendix B. (cid:3) Proof of Theorem 5.1. Let us focus on the evolution of H t via H − t ∂ t H t = ω t ∧ A κt ω t . (43)We first show that there exists e T > such that H t holds diagonal for any t ∈ [0 , e T ) . To this end,it is enough to prove that ω t ∧ ( A κt ) i ¯ j = 0 for any i = j and t = 0 . Thus, let H and ω be twoleft-invariant diagonal Hermitian metrics on G given by H = i (cid:16) ˜ r ζ + ˜ s ζ + ˜ k ζ (cid:17) , ˜ s , ˜ r , ˜ k > , and ω = i (cid:16) r ζ + s ζ + k ζ (cid:17) , s , r , k > . If we consider { e , . . . , e } a left-invariant coframe on G such that δ ζ = e + i e = e − i J e , δ ζ = e + i e = e − i J e , δ ζ = e + i e = e − i J e , with δ = r , δ = s and δ = k , then we get ( A κ ) i ¯ j = 1 δ i δ j (cid:16) ( A κ ) e i e j + i ( A κ ) e i Je j − i ( A κ ) Je i e j + ( A κ ) Je i Je j (cid:17) , HE ANOMALY FLOW ON NILMANIFOLDS 22 where ( A κ ) e i e j are the curvature 2-forms of ∇ κ explicitly computed in Appendix B (see the proof ofLemma 7.1). Thus, the only non-zero entries in the right-hand side of (43) are given by ω ∧ ( A κ ) ω = 112 ˜ k r s k ˜ r h r ˜ k (cid:0) ( κ + 1) s − ρ ( κ − r (cid:1) − k ˜ r ( κ − xr + s ) i ,ω ∧ ( A κ ) ω = 112 ˜ k r s k ˜ s h s ˜ k (cid:0) ( κ + 1) ( x + y ) r − ρ ( κ − s (cid:1) − k ˜ s ( κ − (cid:0) ( x + y ) r + x s (cid:1) i ,ω ∧ ( A κ ) ω = 112 ˜ k r s k ˜ r ˜ s h ρ ( κ − (cid:0) r ˜ s + s ˜ r (cid:1) − ( κ + 1) r s (cid:0) ( x + y )˜ r + ˜ s (cid:1) i , (44)and hence our claim follows, since ω and H are both diagonal.On the other hand, by means of Lemma 5.2 and Corollary 3.8, there also exists b T > such that ω t holds diagonal for any t ∈ [0 , b T ) . Therefore, by the existence of b T > and e T > , it follows that ω t and H t hold diagonal for any t along the flow. Finally, the evolution equations in (42) are a directconsequence of (44), taking into account that s ( t ) = c c r ( t ) and k ( t ) = c c by Corollary 3.8. (cid:3) Remark 5.3. Under the assumptions of Theorem 5.1, we have Tr( A κt ∧ A κt ) = C t ζ , where C t = C t ( ρ, x, y ; ω t , H t ; κ ) is a one-parameter function depending both on the Hermitianstructures and the connection ∇ κ . Remark 5.4. Theorem 5.1 applies to the following Lie groups N k in Table 1 and complex structuresin (41): (1) ρ = 0 , y = 1 and x ∈ R , the Lie group is N ; (2) ρ = y = 0 and x = ± , the Lie groupis N ; (3) ρ = 1 , y ≥ and x > y , the Lie group is N ; (4) ρ = x = y = 0 , the Lie group is N . By [4], this is a classification of all the complex structures in (41). Regarding the existence ofbalanced Hermitian metrics, the list reduces to: • ρ = y = 0 , x = − , the Lie group is N ; • ρ = 1 , y = 0 and x ∈ ( − / , , the Lie group is N .Our next result shows that if the initial metric ω is balanced, then there always exists a connec-tion ∇ κ such that (42) only admits constant solutions. Proposition 5.5. Under the hypotheses of Theorem 5.1, if the initial metric ω is balanced, thenthere exists a Gauduchon connection ∇ κ for which the right-hand side of the system (42) identicallyvanishes, and the only admissible solutions to the Anomaly flow (1) are those with constant H t , i.e H t ≡ H .Proof. As we already showed in the proof of Theorem 3.4, a diagonal metric ω is balanced if andonly if c ( x + i y ) + c = 0 , which is equivalent to require x = − c c and y = 0 , (45) HE ANOMALY FLOW ON NILMANIFOLDS 23 with c = √ r k s > and c = √ s k r > by Corollary 3.8. Therefore, by means of (45), thesystem (42) can be written as ddt ˜ r ( t ) = 23 c c (cid:16) c ( κ + 1) − c ρ ( κ − (cid:17) ˜ k ( t ) r ( t ) ˜ r ( t ) ,ddt ˜ s ( t ) = 23 c (cid:16) c ( κ + 1) − c ρ ( κ − (cid:17) ˜ k ( t ) r ( t ) ˜ s ( t ) ,ddt ˜ k ( t ) = 23 c c (cid:16) c ρ ( κ − − c ( κ + 1) (cid:17) (cid:0) c ˜ s ( t ) + c ˜ r ( t ) (cid:1) ˜ k ( t ) r ( t ) ˜ r ( t ) ˜ s ( t ) . Then, the right-hand side of the system identically vanishes if and only if c ( κ + 1) − c ρ ( κ − = ( c − ρ c ) κ + 2( c + ρ c ) κ + c − ρ c = 0 . (46)Finally, since the discriminant of this quadratic polynomial is given by ∆ = 32 ρc c ≥ , the claim follows. (cid:3) Remarkably, by the proof of Proposition 5.5, we can distinguish the following two remarkablecases: • If c = ρ c , then the only solution to the polynomial (46) is κ = 0 . Therefore, ∇ κ is givenby the Lichnerowicz connection ∇ . • If c = ρ c , then the solutions to (46) are either the Bismut connection ( κ = − ) when ρ = 0 , or the Gauduchon connections ∇ κ ± corresponding to the values κ ± = c + c ± √ c c c − c when ρ = 1 . Remark 5.6. Given a solution H t to (42), it may happen that its Gauduchon connection ∇ κt doesnot satisfy the condition ( A κt ) , = ( A κt ) , = 0 . For instance, let us consider the Lie group arisingfrom (41) when ρ = 1 , x = − and y = 0 , which corresponds to N (see Remark 5.4). Then, thediagonal metric ω = i (cid:16) ζ + ζ + ζ (cid:17) on N is balanced and, by means of Proposition 5.5, onegets that for κ ± = ± √ the system (42) is solved by the constant metric H t ≡ i (cid:16) ζ + ζ + ζ (cid:17) .Nonetheless, the Gauduchon connections ∇ κ ± do not satisfy the condition ( A κ ± t ) , = ( A κ ± t ) , = 0 .Indeed, by Appendix B (see the proof of Lemma 7.1) one gets that ( A κ ± ) has non-zero componentin e + e = − i √ ( ζ − ζ ¯1¯2 ) , and hence its (2 , and (0 , components do not identically vanish.In the following section we prove that, on the Lie group N , solutions to the Hull-Strominger-Ivanov system can obtained as stationary points to the Anomaly flow.5.1. Anomaly flow on N and solutions to the Hull-Strominger-Ivanov system. Let us consider the simply-connected nilpotent Lie group N , which admits a left-invariant (1 , -coframe { ζ , ζ , ζ } satisfying the structure equations ( dζ = dζ = 0 ,dζ = ζ − ζ . (47)Next we study the Anomaly flow (1) on N for ∇ κ being the Chern connection (i.e. κ = 1 ) andthe Strominger-Bismut connection (i.e. κ = − ). HE ANOMALY FLOW ON NILMANIFOLDS 24 • The Chern connection on T , N . We start investigating the setting of Theorem 5.1 in the special case when κ = 1 , i.e. ∇ κt is theChern connection on ( T , N , H t ) . Theorem 5.7. If κ = 1 , then the coefficients of ω t and H t evolve via the ODEs system ddt r ( t ) = c c + α ′ (1 − τ )( τ − τ + 5) c ( c + c )2 r ( t ) ,ddt ˜ r ( t ) = 83 c c ˜ k ( t ) r ( t ) ˜ r ( t ) ,ddt ˜ s ( t ) = 83 c c ˜ k ( t ) r ( t ) ˜ s ( t ) ,ddt ˜ k ( t ) = − c c (cid:16) ˜ r ( t ) + ˜ s ( t ) (cid:17) ˜ k ( t ) r ( t ) ˜ r ( t ) ˜ s ( t ) . (48) Moreover, if ω and H are both balanced, then H t evolves as H t = i r ( t ) ζ + i r ( t ) ζ + i r ˜ k ˜ r ( t ) ζ , where the function ˜ r ( t ) satisfies ddt ˜ r ( t ) = 83 c ˜ r (0) ˜ k (0) r ( t ) ˜ r ( t ) (49) In particular, if τ = 1 (i.e. ∇ τt is different from the Chern connection), then there exists a convenientchoice of α ′ such that the solution to the system is given by ω t ≡ ω and ˜ r ( t ) = √ A t + B , with A = 16 ˜ r ˜ k c r and B = ˜ r .Proof. By means of Proposition 3.3, the first equation of the Anomaly flow (1) reduces to ddt r ( t ) = c K ( t, α ′ , τ ) , (50)where K ( t, α ′ , τ ) is given by K ( t, α ′ , τ ) ζ = i∂∂ω t − α ′ (cid:0) Tr( Rm τt ∧ Rm τt ) − Tr( A t ∧ A t ) (cid:1) . By Corollary 3.8 and Proposition 2.3, a direct computation yields that i∂∂ω t = c c ζ , Tr( Rm τt ∧ Rm τt ) = ( τ − τ − τ + 5) c + c c r ( t ) ζ , while, by means of Lemma 7.1, for κ = 1 we have Tr( A t ∧ A t ) = 0 . Therefore, by using (50) and(42) for κ = 1 , ρ = 0 = y and x = − , one gets the ODEs system (48).Now, let ω and H be both balanced. By means of (35) and (30)–(34), the balanced conditionimplies that c = c and ˜ s = ˜ r . HE ANOMALY FLOW ON NILMANIFOLDS 25 The latter equality, together with the fact that the functions ˜ r ( t ) and ˜ s ( t ) satisfy similar equationsin (48), leads to ˜ s ( t ) = ˜ r ( t ) . Thus, the ODEs system (48) reduces to ddt r ( t ) = c + α ′ (1 − τ )( τ − τ + 5) c r ( t ) ,ddt ˜ r ( t ) = 83 c ˜ k ( t ) r ( t ) ˜ r ( t ) ,ddt ˜ k ( t ) = − c ˜ k ( t ) r ( t ) ˜ r ( t ) . (51)Therefore, by considering the quotient of ddt ˜ r ( t ) with ddt ˜ k ( t ) , we get Z r ( t ) d˜ r ( t ) = − Z k ( t ) d˜ k ( t ) . This in turn implies ˜ k ( t ) = ˜ r ˜ k ˜ r ( t ) , and hence (49) follows.Finally, for any value of r and τ = 1 , there exists a convenient value of α ′ making the right-handside of the first equation in (51) equal to zero. In this case we can explicitly solve the system with ˜ r ( t ) = √ A t + B, where A = 16 ˜ r ˜ k c r and B = ˜ r . (cid:3) We stress that the explicit solutions found in Theorem 5.7 are not stationary solutions to the flow,and hence they do not solve the Hull-Strominger system. In the next subsection, we will constructstationary solutions assuming ∇ κt to be the Strominger-Bismut connection. • The Bismut connection on T , N . Here we consider the setting of Theorem 5.1 in the special case when κ = − , i.e. ∇ κt is theStrominger-Bismut connection on ( T , N , H t ) . Theorem 5.8. If κ = − , then the coefficients of ω t and H t evolve via the ODEs system ddt r ( t ) = c c + α ′ (1 − τ )( τ − τ + 5) c ( c + c )8 r ( t ) − α ′ c k ( t ) (˜ r ( t ) + ˜ s ( t ) )˜ r ( t ) ˜ s ( t ) ,ddt ˜ r ( t ) = 23 c ( c − c ) ˜ k ( t ) r ( t ) ,ddt ˜ s ( t ) = 23 c ( c − c ) ˜ k ( t ) r ( t ) ,ddt ˜ k ( t ) = 0 . (52) If the initial metric ω is balanced, then H t ≡ H is constant, the Strominger-Bismut connection ∇ − of H is a (non-flat) instanton with respect to ω t , and the Anomaly flow reduces to the ODE ddt r ( t ) = K + K r ( t ) , (53) HE ANOMALY FLOW ON NILMANIFOLDS 26 where K = K ( ω , α ′ , H ) and K = K ( ω , α ′ , τ ) are given by K := c − α ′ c k (˜ r + ˜ s )˜ r ˜ s , K := α ′ (1 − τ )( τ − τ + 5) c . (54) Therefore, starting from any balanced initial metric ω on N , we have the following: (i) Given α ′ = 0 and τ ∈ R , the metric H can be conveniently chosen in order to obtain K < , = 0 or > in (53) , and so there always exists a stationary point to the Anomalyflow which solves the Hull-Strominger system. (ii) Furthermore, if α ′ = 0 and τ = − (i.e. ∇ τt is the Strominger-Bismut connection of ω t ),then ∇ − t is an instanton with respect to ω t , and hence there exists a stationary point tothe Anomaly flow which solves the Hull-Strominger-Ivanov system.Proof. The first part of the statement follows the same argument of Theorem 5.7. We just mentionthat, by means of Lemma 7.1 for κ = − , ρ = 0 = y and x = − , we have Tr( A − t ∧ A − t ) = − r ( t ) + ˜ s ( t ) ˜ r ( t ) ˜ s ( t ) ˜ k ( t ) ζ . (55)Hence, the ODEs system (52) is obtained starting from (42).Now, let us assume ω balanced. By means of (35) and (30)–(34), we have c = c , and hence the ODEs system (52) reduces to ˜ r ( t ) , ˜ s ( t ) , ˜ k ( t ) constant (i.e. H t ≡ H ), and ddt r ( t ) = K + K r ( t ) , with K and K as given in (54). Therefore, we get that ω t ∧ A − = 0 for any t ∈ ( T − , T + ) and,by means of the curvature forms given in the proof of Lemma 7.1, a direct computation yields thatthe curvature of the Strominger-Bismut connection satisfies ( A − ) , = ( A − ) , = 0 . (56)Hence, ∇ − is an instanton with respect to ω t for any t ∈ ( T − , T + ) . It is non-flat because Tr( A − ∧ A − ) = 0 by (55).Finally, the last two claims are consequences of (54), (56) and similar arguments to those inSection 4. In greater detail, given α ′ = 0 and τ ∈ R , we can choose a metric H such that K hasopposite sign to that of K (notice that if K = 0 then we can take H so that K also vanishes).Now, (i) follows from a qualitative analysis similar to that given in Section 4. For the proof of(ii), it only remains to prove that the Strominger-Bismut connection ∇ − t of the metric ω t is aninstanton with respect to ω t . This follows from Appendix A for τ = − , ρ = λ = y = 0 , x = − and u = 0 (because the metric ω t remains diagonal). Indeed, in this case it is direct to check that ( Rm − t ) , = ( Rm − t ) , = 0 , so there exists a stationary point to the Anomaly flow which solvesthe system given by (2) and (5), i.e. the Hull-Strominger-Ivanov system. (cid:3) By Theorem 5.8 (ii), when both ∇ τt and ∇ κt are Strominger-Bismut connections and the initialmetric ω is balanced, the Anomaly flow (1) always converges to a solution of the Hull-Strominger-Ivanov system. We note that explicit solutions of this kind were previously found in [13, Theorem5.1] and in [28, Theorem 3.3] by means of other methods. HE ANOMALY FLOW ON NILMANIFOLDS 27 Appendix A In this Appendix we provide the curvature forms for a connection ∇ τ in the Gauduchon familygiven any left-invariant metric ω . We will denote the curvature by Ω τ instead of Rm τ .For the computation of the curvature forms we use (22) with respect to the adapted basis { e l } l =1 found in Proposition 2.1, together with the connection 1-forms ( σ τ ) ij obtained in Proposition 2.3.We first notice that the 2-forms (Ω τ ) ij satisfy the following relations: (Ω τ ) = − (Ω τ ) , (Ω τ ) = (Ω τ ) , (Ω τ ) = − (Ω τ ) , (Ω τ ) = (Ω τ ) , (Ω τ ) = − (Ω τ ) , (Ω τ ) = (Ω τ ) . Next, we give the explicit expression of the 2-forms r e ∆ e k e (Ω τ ) ij , where ∆ e := p r e s e − | u e | , for ( i, j ) = { (1 , , (1 , , (1 , , (1 , , (1 , , (3 , , (3 , , (3 , , (5 , } : r e ∆ e k e (Ω τ ) = − ( τ − τ +5)∆ e e + ( τ − τ +5) u e ∆ e ( e + e )+ (cid:0) ( τ − τ +5) u e − (cid:0) ρ ( τ − τ +3)+ λ ( τ +3) (cid:1) r e (cid:1) ∆ e e − (cid:0) ( τ − τ +5) u e + (cid:0) ρ ( τ − τ +3) − λ ( τ +3) (cid:1) r e (cid:1) ∆ e e − (cid:0) ( τ − τ +5) | u e | − λ ( τ +3) u e r e − ( ρ ( τ − − λ ( τ +1) +4 x ( τ − r e (cid:1) e − λ ( τ − (cid:0) λr e − u e (cid:1) r e e , r e ∆ e k e (Ω τ ) = ( τ − τ +5) u e ∆ e e − (cid:2) ( τ − τ +5) u e + (cid:0) ρ ( τ − − λ ( τ − − ρλ ( τ − τ +3)+ x ( τ +1) (cid:1) r e (cid:3) e − (cid:2) ( τ − τ +5) u e u e − (cid:0) ρ ( τ − τ +3)+ λ ( τ +3) (cid:1) u e r e + y ( τ +1) r e (cid:3) e + (cid:2) ( τ − τ +5) u e u e + (cid:0) ρ ( τ − τ +3) − λ ( τ +3) (cid:1) u e r e + y ( τ +1) r e (cid:3) e − (cid:2) ( τ − τ +5) u e + (cid:0) ρ ( τ − − λ ( τ − ρλ ( τ − τ +3)+ x ( τ +1) (cid:1) r e (cid:3) e + e (cid:2) ( τ − τ +5) | u e | u e − λ ( τ +3) u e u e r e + (cid:0) λ ( τ +3) u e + x ( τ − τ +5) u e + y ( τ +1) u e (cid:1) r e − λy ( τ +3) r e (cid:3) e + e ( τ − (cid:0) λr e − u e (cid:1)(cid:0) λu e − yr e (cid:1) r e e , r e ∆ e k e (Ω τ ) = (cid:0) ( τ − τ +5) u e +2 λ ( τ − r e (cid:1) ∆ e e − (cid:2) ( τ − τ +5) u e u e +2 λ ( τ − u e r e − y ( τ +1) r e (cid:3) ( e + e ) − (cid:2) ( τ − τ +5) u e − (cid:0) ρ ( τ − τ +3)+ λ ( τ − τ +5) (cid:1) u e r e + (cid:0) ρ ( τ − − λ ( τ − ρλ ( τ − τ +3)+ x ( τ +1) (cid:1) r e (cid:3) e + (cid:2) ( τ − τ +5) u e + (cid:0) ρ ( τ − τ +3) − λ ( τ − τ +5) (cid:1) u e r e + (cid:0) ρ ( τ − − λ ( τ − − ρλ ( τ − τ +3)+ x ( τ +1) (cid:1) r e (cid:3) e + e (cid:2) ( τ − τ +5) | u e | u e +2 λ (cid:0) ( τ − u e − ( τ − τ +4) u e (cid:1) r e + (cid:0) λ ( τ +3) u e + x ( τ − τ +5) u e − y ( τ +1) u e (cid:1) r e − λx ( τ +3) r e (cid:3) e − e ( τ − (cid:0) λu e − (cid:0) λs e + λ u e − yu e (cid:1) r e + λxr e (cid:1) r e e , HE ANOMALY FLOW ON NILMANIFOLDS 28 r e ∆ e k e (Ω τ ) = − λ ( τ − (cid:0) ( τ +1) u e − ρ ( τ − r e (cid:1) r e e + ( τ − (cid:0) ρ ( τ − − λ ( τ +1) (cid:1) u e r e e − λ ( τ − τ +1) u e r e e − ( τ − (cid:0) τ +1) s e +2 ρ ( τ − u e − λ ( τ +1) u e − ρλ ( τ − r e (cid:1) r e e + λ e ( τ − τ +1) (cid:0) | u e | − λu e r e + xr e (cid:1) r e e + e ( τ − τ +1) (cid:0) u e s e − ( λ u e − xu e +2 yu e ) r e + λyr e (cid:1) r e e − λ e ( τ − τ +1) (cid:0) λu e − yr e (cid:1) r e e + e ( τ − (cid:2) ρ ( τ − | u e | +( τ +1)(2 s e u e − λ | u e | ) − (cid:0) ρ ( τ − s e +( τ +1)(2 λs e − λ u e − xu e − yu e ) (cid:1) r e − λx ( τ +1) r e (cid:3) r e e , r e ∆ e k e (Ω τ ) = − λ ( τ − τ +1) u e r e e − ( τ − (cid:0) τ +1) s e − ρ ( τ − u e − λ ( τ +1) u e + ρλ ( τ − r e (cid:1) r e e + λ ( τ − (cid:0) ( τ +1) u e + ρ ( τ − r e (cid:1) r e e + ( τ − (cid:0) ρ ( τ − λ ( τ +1) (cid:1) u e r e e − λ e ( τ − τ +1) (cid:0) λu e − yr e (cid:1) r e e − e ( τ − (cid:2) ρ ( τ − | u e | − ( τ +1)(2 s e u e − λ | u e | ) − (cid:0) ρ ( τ − s e − ( τ +1)(2 λs e − λ u e − xu e − yu e ) (cid:1) r e + λx ( τ +1) r e (cid:3) r e e − λ e ( τ − τ +1) (cid:0) | u e | − λu e r e + xr e (cid:1) r e e − e ( τ − τ +1) (cid:0) u e s e − ( λ u e − xu e +2 yu e ) r e + λyr e (cid:1) r e e , r e ∆ e k e (Ω τ ) = − (cid:0) ( τ − τ +5) | u e | +4 λ ( τ − u e r e − ( τ − ρ ( τ − x ) r e (cid:1) e + e (cid:2) ( τ − τ +5) | u e | u e +4 λ ( τ − u e u e r e − (cid:0) λ ( τ − u e + ρλ ( τ − τ +3) u e − x ( τ − τ +5) u e + y ( τ +1) u e (cid:1) r e + y ( τ − ρ ( τ +3)+2 λ ) r e (cid:3) e + e (cid:2) ( τ − τ +5) | u e | u e − (cid:0) ρ ( τ − τ +3) | u e | + λ ( τ +3) u e + λ ( τ − τ +7) u e (cid:1) r e − (cid:0) λ ( τ − u e − ρλ ( τ − τ +3) u e − x ( τ − τ +5) u e − y ( τ +1) u e (cid:1) r e − x ( τ − ρ ( τ +3) − λ ) r e (cid:3) e − e (cid:2) ( τ − τ +5) | u e | u e + (cid:0) ρ ( τ − τ +3) | u e | − λ ( τ +3) u e − λ ( τ − τ +7) u e (cid:1) r e − (cid:0) λ ( τ − u e + ρλ ( τ − τ +3) u e − x ( τ − τ +5) u e − y ( τ +1) u e (cid:1) r e + x ( τ − ρ ( τ +3)+2 λ ) r e (cid:3) e + e (cid:2) ( τ − τ +5) | u e | u e +4 λ ( τ − u e u e r e − (cid:0) λ ( τ − u e − ρλ ( τ − τ +3) u e − x ( τ − τ +5) u e + y ( τ +1) u e (cid:1) r e − y ( τ − ρ ( τ +3) − λ ) r e (cid:3) e − τ − τ +5∆2 e (cid:2) | u e | − λ | u e | u e r e +(2 x + λ ) | u e | r e − λ ( xu e + yu e ) r e +( x + y ) r e (cid:3) e λ ( τ − ( λr e − u e ) r e e , HE ANOMALY FLOW ON NILMANIFOLDS 29 r e ∆ e k e (Ω τ ) = − e ( τ − τ +1) (cid:0) λ | u e | +( λs e − yu e ) r e (cid:1) r e e + e ( τ − τ +1) (cid:0) λu e − s e − xr e (cid:1) u e r e e + e ( τ − λu e − yr e ) (cid:0) ( τ +1) u e + ρ ( τ − r e (cid:1) r e e + e ( τ − (cid:2) ρ ( τ − | u e | + λ ( τ +1)( u e − u e )+2( τ +1) s e u e − (cid:0) ρλ ( τ − u e + λ ( τ +1) s e − x ( τ +1) u e (cid:1) r e +2 ρx ( τ − r e (cid:3) r e e − e ( τ − (cid:2) λ ( τ +1) | u e | u e + (cid:0) ρλ ( τ − | u e | + λ ( τ +1)( u e − u e ) − λ ( τ +1) s e u e (cid:1) r e − (cid:0) ρλs e ( τ − − λ ( τ +1)( λs e − yu e ) (cid:1) r e +2 y ( τ +1) r e (cid:3) r e e − e ( τ − (cid:2) (2 ρ ( τ − λ ( τ +1)) | u e | u e − (cid:0) ρ ( τ − s e u e − ( τ +1)( λs e u e − λ u e u e − y | u e | ) (cid:1) r e +2 λ ( τ +1)( xu e + yu e ) r e − xy ( τ +1) r e (cid:3) r e e − e ( τ − τ +1) (cid:2) λ | u e | u e +( λs e u e − λ u e u e − y | u e | ) r e +2 λ ( xu e + yu e ) r e − xyr e (cid:3) r e e − e ( τ − (cid:2)(cid:0) ρ ( τ − u e − ( τ +1)( λu e − s e ) (cid:1) | u e | + (cid:0) ρλ ( τ − s e + λ ( τ +1)( λs e − xu e ) (cid:1) r e +2 x ( τ +1) r e − (cid:0) ρ ( τ − λ | u e | +2 s e u e )+( τ +1)(3 λs e u e + λ ( u e − u e ) − x | u e | ) (cid:1) r e (cid:3) r e e , r e ∆ e k e (Ω τ ) = e ( τ − λu e − yr e ) (cid:0) ( τ +1) u e − ρ ( τ − r e (cid:1) r e e − e ( τ − (cid:2) ρ ( τ − | u e | − ( τ +1)( λ ( u e − u e )+2 s e u e ) − (cid:0) ρλ ( τ − u e − λ ( τ +1) s e +2 x ( τ +1) u e (cid:1) r e +2 ρx ( τ − r e (cid:3) r e e + e ( τ − τ +1) (cid:2) λ ( u e − u e )+( λs e − yu e ) r e (cid:3) r e e − e ( τ − τ +1) (cid:2) λu e − s e − xr e (cid:3) u e r e e − e ( τ − (cid:2)(cid:0) τ +1) s e − (2 ρ ( τ − λ ( τ +1)) u e (cid:1) | u e | − (cid:0) ρλ ( τ − s e − λ ( τ +1)( λs e − xu e ) (cid:1) r e +2 x ( τ +1) r e + (cid:0) ρ ( τ − s e u e + λ | u e | ) − ( τ +1)(3 λs e u e + λ ( u e − u e ) − x | u e | ) (cid:1) r e (cid:3) r e e + e ( τ − (cid:2) λ ( τ +1) u e | u e | − (cid:0) ρλ ( τ − | u e | − λ ( τ +1)( λ ( u e − u e ) − s e u e ) (cid:1) r e + (cid:0) ρλ ( τ − s e + λ ( τ +1)( λs e − yu e ) (cid:1) r e +2 y ( τ +1) r e (cid:3) r e e − e ( τ − (cid:2) (2 ρ ( τ − − λ ( τ +1)) u e | u e | − λ ( τ +1)( xu e + yu e ) r e +2 xy ( τ +1) r e − (cid:0) ρ ( τ − s e u e + λ ( τ +1) s e u e − τ +1)( λ u e u e + y | u e | ) (cid:1) r e (cid:3) r e e , r e ∆ e k e (Ω τ ) = − r e ∆ e k e (cid:2) (Ω τ ) + (Ω τ ) (cid:3) − τ − λu e − s e − xr e ) r e e + e ( τ − (cid:0) u e ( λu e − s e ) − ( ρλ + λ +2 x ) u e r e +( ρ + λ ) yr e (cid:1) r e e + e ( τ − (cid:0) u e +( ρ − λ ) r e (cid:1)(cid:0) λu e − s e − xr e (cid:1) r e e − e ( τ − (cid:0) u e − ( ρ + λ ) r e (cid:1)(cid:0) λu e − s e − xr e (cid:1) r e e + e ( τ − (cid:0) u e ( λu e − s e )+( ρλ − λ − x ) u e r e − ( ρ − λ ) yr e (cid:1) r e e − e ( τ − (cid:2) ( λu e − s e ) | u e | + (cid:0) λu e s e − λ | u e | − x | u e | (cid:1) r e + (cid:0) λ ( xu e + yu e ) − xs e (cid:1) r e − ( x + y ) r e (cid:3) r e e . Appendix B This Appendix is devoted to the computation of Tr( A κ ∧ A κ ) for a Gauduchon connection ∇ κ onthe holomorphic tangent bundle T , G . In particular, the proof of Lemma 5.2 will follow. HE ANOMALY FLOW ON NILMANIFOLDS 30 Let ( G, J ) be a Lie group equipped with a left-invariant complex structure. Let { ζ , ζ , ζ } bea left-invariant (1,0)-coframe satisfying (41). Let also ω and H be two left-invariant J -Hermitianmetrics on G given by ω = i r ζ + s ζ + k ζ ) and H = i r ζ + ˜ s ζ + ˜ k ζ ) , (57)for some r, s, k, ˜ r, ˜ s, ˜ k ∈ R ∗ . If ∇ κ is a Gauduchon connection of H and A κ its curvature form, tocompute the trace Tr( A κ ∧ A κ ) by using (22) and (23) we need to write the connection 1-forms σ κ in terms of an adapted basis { e l } l =1 for ω (see Proposition 2.1). In the following, we will denote by ( σ κ ) ij and ( A κ ) ij the connection 1-forms and the curvature 2-forms, respectively, written in terms of { e l } l =1 .Let { ˜ e l } l =1 be an adapted basis for the metric H and { ˜ e l } l =1 its dual. In view of Section 2.2, theconnection 1-forms ( σ κ ) ˜ i ˜ j associated to ∇ κ are given by ∇ ˜ e k ˜ e j = ( σ κ ) ˜1˜ j (˜ e k ) ˜ e + · · · + ( σ κ ) ˜6˜ j (˜ e k ) ˜ e . On the other hand, if { e l } l =1 denotes the dual basis of { e l } l =1 , and M := ( M pj ) is the change-of-basismatrix from { e l } to { ˜ e l } , i.e. ˜ e j = M pj e p , for every ≤ j ≤ , then one gets ∇ ˜ e k ˜ e j = ∇ M pk e p ( M qj e q ) = M pk M qj ∇ e p e q = M pk M qj ( σ κ ) lq ( e p ) e l = M pk M qj N il ( σ κ ) lq ( e p )˜ e i , with N := M − (that is, e l = N il ˜ e i ), and hence ( σ κ ) ˜ i ˜ j (˜ e k ) = ˜ g ( ∇ ˜ e k ˜ e j , ˜ e i ) = M pk M qj N il ( σ κ ) lq ( e p ) . (58)Since the (1 , -coframe { ζ , ζ , ζ } only depends on the complex structure J , by means of (14),(18) and (20) we have e + i e = r ζ , ˜ r ζ = ˜ e + i ˜ e ,e + i e = s ζ , ˜ s ζ = ˜ e + i ˜ e ,e + i e = k ζ , ˜ k ζ = ˜ e + i ˜ e , which directly implies ˜ e = ˜ rr e , ˜ e = ˜ rr e , ˜ e = ˜ ss e , ˜ e = ˜ ss e , ˜ e = ˜ kk e , ˜ e = ˜ kk e . Thereby, the change-of-basis matrix M from { e l } to { ˜ e l } is given by the diagonal matrix M := diag (cid:18) r ˜ r , r ˜ r , s ˜ s , s ˜ s , k ˜ k , k ˜ k (cid:19) . Thus, by means of (58), one gets ( σ κ ) ij ( e k ) = M ii N jj N kk ( σ κ ) ˜ i ˜ j (˜ e k ) , (59)or, equivalently, ( σ κ ) ij = M ii N jj N kk ( σ κ ) ˜ i ˜ j (˜ e k ) e k . HE ANOMALY FLOW ON NILMANIFOLDS 31 Finally, since the connection 1-forms ( σ κ ) ˜ i ˜ j are given in the proof of Proposition 2.3, a direct com-putation by means of (59) yields that ( σ κ ) = − ˜ k k ˜ r ( κ − e , ( σ κ ) = − ˜ k k ˜ r ( κ +1) e + r ˜ k s k ˜ r ρ ( κ − e , ( σ κ ) = ˜ k k ˜ r ( κ +1) e + r ˜ k s k ˜ r ρ ( κ − e , ( σ κ ) = ˜ k k ˜ s y ( κ − e − ˜ k k ˜ s x ( κ − e , ( σ κ ) = − s ˜ k r k ˜ s ρ ( κ − e − ˜ k k ˜ s x ( κ +1) e − ˜ k k ˜ s y ( κ +1) e , ( σ κ ) = − s ˜ k r k ˜ s ρ ( κ − e − ˜ k k ˜ s y ( κ +1) e + ˜ k k ˜ s x ( κ +1) e , ( σ κ ) =( σ κ ) = ( σ κ ) = ( σ κ ) = ( σ κ ) = 0 , (60)together with the following relations ( σ κ ) = − ( σ κ ) , ( σ κ ) = ( σ κ ) , ( σ κ ) = − ( σ κ ) , ( σ κ ) = ( σ κ ) , and ( σ κ ) ij = − ( σ κ ) ji . Lemma 7.1. Let G be a -step nilpotent Lie group equipped with a left-invariant complex structure J which admits a left-invariant (1 , -coframe { ζ l } l =1 satisfying (41) . Let ω and H be two left-invariant J -Hermitian metrics defined by (57) . Then, for any Gauduchon connection ∇ κ associatedto H , the trace of its curvature satisfies Tr( A κ ∧ A κ ) = Cζ , where C = C ( ρ, x, y ; ω, H ; κ ) is a constant depending both on the Hermitian structures and theconnection. More precisely, we have Tr ( A κ ∧ A κ ) = ( κ − 1) ˜ k k ˜ r ˜ s n ρ ( κ − h (2 κ r ˜ k + k ˜ r )˜ s + ( x + y )(2 κ s ˜ k + k ˜ s )˜ r i + 4 x ( κ − (cid:16) ( x + y )˜ r + ˜ s (cid:17) k ˜ r ˜ s − x ( κ + 1) (cid:16) ( x + y ) s ˜ r + r ˜ s (cid:17) ˜ k o ζ . HE ANOMALY FLOW ON NILMANIFOLDS 32 Proof. Let ( σ κ ) ij be the connection 1-forms of the Gauduchon connection ∇ κ given in (60). Bymeans of (11) and (22), a direct computations yields that the curvature 2-forms ( A κ ) ij of ∇ κ are ( A κ ) = κ − k ˜ r ˜ k − ( κ +1) r ˜ k r k ˜ r e − ρ ( κ − ( ( κ +1) r ˜ k +2 k ˜ r ) ˜ k rsk ˜ r ( e + e )+ ( κ − ( ρ ( κ − r ˜ k +4 xk ˜ r ) ˜ k s k ˜ r e , ( A κ ) = − ( ρ ( κ − + x ( κ +1) ) ˜ k k ˜ r ˜ s ( e + e ) − y ( κ +1) ˜ k k ˜ r ˜ s ( e − e ) , ( A κ ) = y ( κ +1) ˜ k k ˜ r ˜ s ( e + e ) − ( ρ ( κ − + x ( κ +1) ) ˜ k k ˜ r ˜ s ( e − e ) , ( A κ ) = − ( κ − κ +1)˜ k k ˜ r e − ρ ( κ − r ˜ k sk ˜ r e , ( A κ ) = − ( κ − κ +1)˜ k k ˜ r e + ρ ( κ − r ˜ k sk ˜ r e , ( A κ ) = ( κ − ( ρ ( κ − s ˜ k +4 xk ˜ s ) ˜ k r k ˜ s e + ρ y ( κ − ( ( κ +1) s ˜ k +2 k ˜ s ) ˜ k rsk ˜ s ( e − e ) − ρ x ( κ − ( ( κ +1) s ˜ k +2 k ˜ s ) ˜ k rsk ˜ s ( e + e ) + ( x + y ) ( κ − k ˜ s − ( κ +1) s ˜ k ) ˜ k s k ˜ s e , ( A κ ) = − ρ ( κ − s ˜ k rk ˜ s ( y e − x e ) − ( κ − κ +1)˜ k k ˜ s (cid:0) y e − xy ( e + e ) + x e (cid:1) , ( A κ ) = ρ ( κ − s ˜ k rk ˜ s ( y e − x e ) + ( κ − κ +1)˜ k k ˜ s (cid:0) xy ( e − e ) − x e + y e (cid:1) , ( A κ ) = − ( ρ ( κ − s ˜ r − ( κ +1) r ˜ s ) ˜ k r k ˜ r ˜ s e − ρy ( κ − κ +1) s ˜ k rk ˜ s ( e − e )+ ρ ( κ − κ +1)( r ˜ s + x s ˜ r )˜ k rsk ˜ r ˜ s ( e + e ) − ( ρ ( κ − r ˜ s − ( x + y )( κ +1) s ˜ r ) ˜ k s k ˜ r ˜ s e , together with the following relations ( A κ ) = − ( A κ ) , ( A κ ) = ( A κ ) , ( A κ ) = − ( A κ ) , ( A κ ) = ( A κ ) , ( A κ ) = − ( A κ ) , ( A κ ) = ( A κ ) . 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